My Scene bpy.types.Panel appears on all tabs
It looks like I forgot to set up the bl_context member of my panel. Sadly, the documentation is not very clear about the possible values (it would be great it if someone could list them and link it in a comment).

For the scene tab, the panel code would be:

class MK8PanelSceneCourse(bpy.types.Panel):
    bl_label = "Mario Kart 8 Course"
    bl_idname = "SCENE_PT_mk8course"
    bl_space_type = "PROPERTIES"
    bl_region_type = "WINDOW"
    bl_context = "scene"